1Department of Radiology and Biomedical Imaging, University of California San Francisco, San Francisco, CA, USA; 2Magwale, Palo Alto, CA, USA; 3UCSF/UC Berkeley Joint Group Program in Bioengineering, San Francisco and Berkeley, CA, USA

Abstract: Low resolution images are often acquired in in vivo MR applications involving in large field-of-view (FOV) and high speed imaging, such as, whole-body MRI screening and functional MRI applications. In this work, we investigate a multi-slice imaging strategy for acquiring low resolution images by using compressed sensing (CS) MRI to enhance the image quality without increasing the acquisition time. In this strategy, low resolution images of all the slices are acquired using multiple-slice imaging sequence. In addition, extra randomly sampled data in one center slice are acquired by using the CS strategy. These additional randomly sampled data are multiplied by the weighting functions generated from low resolution full k-space images of the two slices, and then interpolated into the k-space of other slices. In vivo MR images of human brain were employed to investigate the feasibility and the performance of the proposed method. Quantitative comparison between the conventional low resolution images and those from the proposed method was also performed to demonstrate the advantage of the method.

Introduction

In in vivo MR imaging applications such as whole-body screening (1-11) and activation mapping in functional MRI (12-25), relatively low resolution MR imaging is often used due to the requirement of large field-of-view (FOV) or rapid imaging speed. The tradeoff for the shortened acquisition time in low resolution imaging is the dramatically reduced acquisition of high frequency components of MR signals. Images with lack of high frequency information provide limited details of the imaging objects. Recently, compressed sensing (CS) imaging technique (26,27) has been used to reduce the acquisition time and raw data size by significantly undersampling the k-space for MRI (28-50). Based on CS technique, the interpolated compressed sensing (iCS) MRI (51-53) is proposed and has demonstrated the advantages in multiple slice MR imaging acquisitions for improving image quality and contrast by utilizing the raw from other slices and the weighting functions. In this work, we investigate a novel strategy to improve image quality for multiple-slice low resolution imaging by using the sparse sampling technique. In a multiple-slice low resolution acquisition, on the top of the k-space data acquired for forming the low resolution image, more k-space data of one center slice will be acquired by using the incoherent sampling strategy. The extra k-space data of the center slice would be able to enhance the high frequency information and thus increase the image resolution, ultimately providing more detailed information of the image. Based on the low resolution data, weighting functions which reflect the difference between the center slice and other slices (or target slices) can be generated. The extra data acquired in the center slice will be interpolated to the other slices after multiplying the corresponding weighting functions. The image reconstruction for each slice will then be performed by using the CS reconstruction algorithm. This strategy is able to enhance the quality of the low resolution images in the multiple slice imaging, in terms of resolution, contrast and image fidelity. In vivo MR imaging of human brain was applied to investigate the feasibility and performance of the proposed method. Comparison with original low resolution images in image error was also performed.

Theory and methods

The proposed strategy for multiple-slice acquisition of the low resolution images is shown in Figure 1. To acquire the multiple-slice low resolution images together with the randomly undersampled k-space data of one center slice, we need to modify the conventional imaging sequences by adding sparse acquisition strategy to the center slice.

Figure 1 Flowchart of the procedure for improving the image quality by using sparse undersampled raw data. CG, Conjugate Gradient.

Firstly, the weighting functions between the center slice and the other slices are generated by calculating the quotient between the two images:

[1]

where I1 and I2 denote the original low resolution images of the center slice and the other slices, respectively. By taking Fourier Transform the weighting functions in k-space domain are obtained:

[2]

where Wk is the weighting function in k-space.

Secondly, the estimated k-space data of the target slice are calculated by taking convolution of the weighting function and the undersampled k-space data of the center slice:

[3]

where Sk_center is the raw data of the center slice undersampled by using sparse MRI strategy, while Sk_new is the estimated raw data of the target slice.

The final step is to interpolate these estimated data to the k-space of the original low resolution images of the target slices. By using nonlinear Conjugate Gradient (CG) reconstruction similar to that used in conventional CS MRI, an improved image with improved image resolution and lower image error can be obtained.

To validate the feasibility of the method, an acquisition example, capable of implementing the proposed method with human brain imaging was designed. The design procedure is shown in Figure 2. A total of 9 slices were acquired at low resolution. The extra randomly undersampled data were acquired for the center slice. These acquisitions (i.e., original low resolution and the extra data for the center slice) could be combined in one single imaging sequence and accomplished in a single acquisition. In image reconstruction, the weighting functions were calculated according to Eq. [1] and Eq. [2] and the high frequency k-space data of the other 8 slices were estimated by using the Eq. [3]. Finally, nonlinear CG method was used to perform image construction for all slices by using the estimated k-space data.

Figure 2 Diagram of the multi-slice two dimensional low resolution MR imaging strategy with interpolated compressed sensing undersampling data used in our MR experiment. A total of 9 slices were acquired in a single acquisition sequence. The center slice was undersampled by using both the sparse undersampling strategy (undersampling rate is 1/4 along phase encoding) and the low resolution sampling, while the other 8 slices were acquired using only regular low resolution sampling. The extra raw data of the center slice together with the weighting functions were used to estimate the missing raw data of the other 8 slices. The conjugated gradient algorithm was used to perform image reconstruction for all slices.

Image errors in the reference and undersampled images were calculated to evaluate reconstruction performance. The image errors were obtained by subtracting the reconstructed images from the full k-space reference images. Specifically, the image error calculation used was calculated by using:

[4]

where represents the signal intensity of the jth pixel in the full k-space reference images, and represents the signal intensity of the jth pixel in the undersampled images reconstructed using the proposed method or the low resolution images.

In comparison studies, we performed image reconstruction using other two methods—zero filling method and conventional CS method at the same acquisition time (or the same undersampling rate) as that used in the proposed iCS acquisition. Image error maps and contrast to noise ratio (CNR) were compared to evaluate the performance of the three different methods.

Results

Figure 3 shows the human brain images of all the other 8 slices (except the center slice). The first column illustrates the images reconstructed from full k-space data (raw data size was 512 by 512) which serve as the reference images. The second column is the images reconstructed from the proposed method. The third column is the images reconstructed from the low resolution k-space data with 75 phase encoding steps which has an equivalent undersampling rate to the proposed method in this study. The number of phase encoding for iCS acquisition was 64, while the sparsely undersampling rate of the center slice was 1/4 in phase encoding direction (128 phase encoding). From the results shown in Figure 3, it is obvious that the image quality of the iCS method is higher than those of the same slice at low resolution acquisition. For quantitative evaluation and comparison, Eq. [4] was used to calculate the image errors for all slices. The residual images are show in Figure 4. The image errors for all the slices are shown in Table 1. These results demonstrate the significantly improved image fidelity of the proposed method.

Table 1 The average image error of the iCS reconstructed image and the conventional zero filling reconstructed images compared with the full k-space reference image. By using the proposed iCS method, the image error of each slice can be reduced significantly compared with that of the zero filling method at the same acquisition timeFull table

Figure 5 shows the CNR of the images shown in Figure 3. The average CNR of each image is shown in Table 2. From the CNR maps and Table 2 we can see that the CNR of the iCS images are much higher than that of the low resolution images at the same acquisition time.

Table 2 The average CNR of the iCS reconstructed image and the conventional zero filling reconstructed images. The CNR of each slice is significantly improved by using the iCS method compared with that of the conventional zero filling reconstructed images at the same acquisition timeFull table

In the comparison with conventional CS method, undersampled images was acquired at the same acquisition time equivalent to that used for the iCS method, that is, the total phase encoding lines was 75 for each of the 9 slices in the multi-slice acquisition. By using the same conjugated gradient reconstruction strategy, conventional CS reconstructed images were obtained. The results of the comparison between the images reconstructed from the iCS method and conventional CS method for all the slices are shown in Figure 6. The average image error of conventional CS method was 14% larger than that using the proposed interpolated CS method. In addition, enlarged artifacts can be clearly observed in the images reconstructed using conventional CS. This further demonstrates the advantage of the proposed strategy in low resolution multiple-slice imaging.

Figure 3 8-slice in vivo human brain MR images in axial plane. The first column shows the reference images fully sampled (512×512 matrix size) and reconstructed using sum of square method. The second column shows the images acquired and reconstructed by using the proposed iCS method. The undersampling rate for the center slice is 1/4 and the low resolution image of all the 9 slices are 1/8. The third column shows the low resolution images with the same raw data size (~1/7 raw data for each slice). iCS, interpolated compressed sensing.

Figure 4 The comparison in the image errors. The first column is the image error of the images acquired by using the proposed iCS method. The second column is the image error of the zero filling reconstructed images. The iCS images show the much reduced average image error over the low resolution images with zero-filling. iCS, interpolated compressed sensing.

Figure 5 Comparison of the contrast to noise ratio (CNR) of the two imaging strategies. The first column is the CNR of iCS images. The second column is the CNR of the zero filling reconstructed images. Comparing with the low resolution images with zero-filling, higher CNR of iCS images can be clearly observed. iCS, interpolated compressed sensing.

Figure 6 Comparison between the images reconstructed from the iCS method (left column) and conventional CS method (right column) at the same acquisition time or the same undersampling rate. The average image error of the images from CS method is 14% higher than that of the images from iCS reconstruction. iCS, interpolated compressed sensing; CS, compressed sensing.

Discussion and conclusions

A method for improving imaging quality in multi-slice low resolution imaging using iCS is proposed and investigated. The promising results in the in vivo human brain imaging validation and the comparison with the conventional imaging method at the same acquisition time have demonstrate the feasibility and advantages of the proposed method for multi-slice low resolution imaging. By acquiring extra sampling data of one center slice using the incoherent undersampling strategy in CS, more high frequency information in the k-space can be obtained in all slices, ultimately leading to improved images with higher CNR and spatial resolution. The proposed technique might directly benefit the imaging applications with the requirement of large FOV and/or fast acquisition.

In the proposed technique, the accuracy of the weighting function is critical to the accuracy of the interpolated k-space data and thus image fidelity. In generation of the weighting functions for interpolating the k-space data into the target slices, due to the finite Fourier transform and the noise generated during the scanning procedure, the weighting functions generated might not be accurate enough.

Acknowledgements

Funding: This work was partially supported by NIH grants EB008699, K99EB015487 and P41 EB013598, a QB3 Research Award, a Springer Med Fund Award, and a National Natural Science Foundation of China Grant (51228702).